 1.5.1E: Refer to the matrices in eq. Find(a) A + B________________(b) A + C...
 1.5.2E: Refer to the matrices in eq. Find(a) B + C________________(b) 3A___...
 1.5.3E: Refer to the matrices in eq. Find a matrix D such that A + D = Beq
 1.5.4E: Refer to the matrices in eq. Find a matrix D such that A + 2D = Ceq
 1.5.5E: Refer to the matrices in eq. Find a matrix D such that A + 2B + 2D ...
 1.5.6E: Refer to the matrices in eq. Find a matrix D such that 2A + 5B + D ...
 1.5.7E: perform the indicated computation, using the vectors in eq(1) and t...
 1.5.8E: perform the indicated computation, using the vectors in eq(1) and t...
 1.5.9E: perform the indicated computation, using the vectors in eq(1) and t...
 1.5.10E: perform the indicated computation, using the vectors in eq(1) and t...
 1.5.11E: perform the indicated computation, using the vectors in eq(1) and t...
 1.5.12E: perform the indicated computation, using the vectors in eq(1) and t...
 1.5.13E: In vectors eq each exercise, find scalars a1 and a2 that satisfy th...
 1.5.14E: In vectors eq each exercise, find scalars a1 and a2 that satisfy th...
 1.5.15E: In vectors eq each exercise, find scalars a1 and a2 that satisfy th...
 1.5.16E: In vectors eq each exercise, find scalars a1 and a2 that satisfy th...
 1.5.17E: In vectors eq each exercise, find scalars a1 and a2 that satisfy th...
 1.5.18E: In vectors eq each exercise, find scalars a1 and a2 that satisfy th...
 1.5.19E: In vectors eq each exercise, find scalars a1 and a2 that satisfy th...
 1.5.20E: In vectors eq each exercise, find scalars a1 and a2 that satisfy th...
 1.5.21E: Refer to the matrices in eq(1) and in vectors eq(2). Find w2, where...
 1.5.22E: Refer to the matrices in eq(1) and in vectors eq(2). Find w2, where...
 1.5.23E: Refer to the matrices in eq(1) and in vectors eq(2). Find w3, where...
 1.5.24E: Refer to the matrices in eq(1) and in vectors eq(2). Find w3, where...
 1.5.25E: The matrices In eq. Find each of the following. (A + B)Ceq
 1.5.26E: The matrices In eq. Find each of the following. (A + 2B)Aeq
 1.5.27E: The matrices In eq. Find each of the following. (A + C)Beq
 1.5.28E: The matrices In eq. Find each of the following. (B + C)Zeq
 1.5.29E: The matrices In eq. Find each of the following. A(BZ)eq
 1.5.30E: The matrices In eq. Find each of the following. Z(AB)eq
 1.5.31E: The matrices and vectors listed in eq Find the each of the followin...
 1.5.32E: Thematrices and vectors listed in eq. Find the each of the followin...
 1.5.33E: The matrices and vectors listed in eq. Find the each of the followi...
 1.5.34E: The matrices and vectors listed in eq. Find the each of the followi...
 1.5.35E: The matrices and vectors listed in eq. Find the each of the followi...
 1.5.36E: The matrices and vectors listed in eq. Find the each of the followi...
 1.5.37E: The matrices and vectors listed in eq. Find the each of the followi...
 1.5.38E: The matrices and vectors listed in eq. Find the each of the followi...
 1.5.39E: The matrices and vectors listed in eq. Find the each of the followi...
 1.5.40E: The matrices and vectors listed in eq. Find the each of the followi...
 1.5.41E: The matrices and vectors listed in eq. Find the each of the followi...
 1.5.42E: the given matrix is the augmented matrix for a system of linear equ...
 1.5.43E: the given matrix is the augmented matrix for a system of linear equ...
 1.5.44E: the given matrix is the augmented matrix for a system of linear equ...
 1.5.45E: the given matrix is the augmented matrix for a system of linear equ...
 1.5.46E: the given matrix is the augmented matrix for a system of linear equ...
 1.5.47E: the given matrix is the augmented matrix for a system of linear equ...
 1.5.48E: the given matrix is the augmented matrix for a system of linear equ...
 1.5.49E: the given matrix is the augmented matrix for a system of linear equ...
 1.5.50E: (AB) u and A(Bu) the calculations (AB) u and A(Bu) produce the same...
 1.5.51E: The next section will show that all the following calculations prod...
 1.5.52E: Refer to the matrices and vectors in Eq.a) Identify the column vect...
 1.5.53E: Determine whether the following matrix products are defined. When t...
 1.5.54E: What is the size of the product (AB)(CD), where A is(2 × 3) and B i...
 1.5.55E: If A is a matrix, what should the symbol A2 mean? What restrictions...
 1.5.56E: Set where b ? 0. Show that O, A, and B are solutions to the matrix ...
 1.5.57E: Two newspapers compete for subscriptions in a region with 300,000 h...
 1.5.58E: Let a) Find all matrices Such that AB = BA________________b) Use th...
 1.5.59E: Let A and B be matrices such that the product AB is defined and is ...
 1.5.60E: Let A and B be matrices such that the product AB is defined. Use Th...
 1.5.61E: a) Express each of the linear systems i) and ii) in the form Ax = b...
 1.5.62E: Solve Ax = b, where A and b are given by
 1.5.63E: (Reference Theorem 6)
 1.5.64E: For A and C, which follow, find a matrix B (if possible) such that ...
 1.5.65E: For A and C, which follow, find a matrix B (if possible) such that ...
 1.5.66E: A (3 × 3) matrix T = (tij) is called an uppertriangular matrix if T...
 1.5.67E: An (m × n) matrix T = (tij) is called upper triangular if tij= 0 wh...
 1.5.68E: find the vector form for the general solution. x1 + 3x2 ? 3x3 + 2x4...
 1.5.69E: find the vector form for the general solution. 14x1 ? 8x2 + 3x3 ? 4...
 1.5.70E: find the vector form for the general solution. 18x1 + 18x2 ? 10x3 +...
Solutions for Chapter 1.5: Introduction to Linear Algebra 5th Edition
Full solutions for Introduction to Linear Algebra  5th Edition
ISBN: 9780201658590
Solutions for Chapter 1.5
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Introduction to Linear Algebra was written by and is associated to the ISBN: 9780201658590. Chapter 1.5 includes 70 full stepbystep solutions. This textbook survival guide was created for the textbook: Introduction to Linear Algebra , edition: 5. Since 70 problems in chapter 1.5 have been answered, more than 7160 students have viewed full stepbystep solutions from this chapter.

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.